CURVES AND MOTION VIA VECTOR VALUED FUNCTIONS A. HAVENS 0. Notes on these Notes These notes cover material spanning several lectures, focusing on curves and functions whose outputs are viewed as position vectors. The notes also contain a few additional remarks on curves and examples that could not be covered in class, as well as some good exercises. A separate, optional, and more challenging set of notes has been made to illuminate the meanings of various geometric objects associated to curves, such as vector and scalar curvature, torsion, and natural frames: see Curvature, Natural Frames, and Acceleration for Plane and Space Curves as linked on the course website. 1. Vector Valued Functions and Curves 1.1. Describing Curves Parametrically. One often first encounters plane curves as graphs of continuous, single variable real-valued functions, and subsequently is introduced to implicit curves. For example, you are probably familiar with the `[' shape of the graphs of functions y = x2n, 2 n = 1; 2 :::, and you have surely encountered the unit circle as the locus of all points (x; y) 2 R satisfying x2 + y2 = 1, which is likely the first implicitly defined curve to become familiar to you. One can also take another approach: introduce a parameter, and describe each of the constituent entries of the ordered pair (x; y) as functions of this common parameter, resulting in a parametric description of a curve. For example, the unit circle can be described by many parameterizations, but perhaps the simplest descriptions are x(θ) = cos θ ; y(θ) = sin θ ; 0 ≤ θ ≤ 2π ; or x(t) = sin(t) ; y(t) = cos(t) ; t 2 R : Exercise 1.1. Describe the differences in these parameterizations from a geometric standpoint (what do θ and t represent?). Then, imagine that the parameterizations describe particle motion, and explain the differences in their motions. One can recover curves which are graphs from a parameterization as well: simply let x(t) = t, and y(t) = f(t), and the resulting curve, for t 2 Dom(f) is simply the graph y = f(x). This parameterization is of course not unique: we can replace t by any function g(t) which is monotonic, and whose range contains Dom(f) and obtain a new parameterization. For example, if Dom(f) = 3 3 R one could also parameterize the graph of y = f(x) by x(t) = t , y(t) = f(t ). Viewing the parameterization as describing motion along the curve, we realize that these parameterizations are quite different in nature, though they describe the same plane curve. Parameterizations have greater freedom to describe curves than graphs, or even implicit func- tions. As a testament to this, consider the cycloid: a cycloid is the shape made by following a point on the circumference of a disk, as the disk rolls without slipping along a straight path. If the disk has radius R and rolls along the x-axis starting at time t = 0, one can deduce that the cycloid has parametric equations x(t) = R(t − sin t) y(t) = R(1 − cos t) 0 ≤ t : One can express a portion of one arch by an equation giving x as a function of y, but no closed form f exists for which y = f(x) describes the whole cycloid. 1 2/21/20 Multivariate Calculus: Curves Havens Figure 1. A pair of moments in a disk's motion, following one point on the cir- cumference as it traces out a cycloid. Exercise 1.2. Use a diagram and basic trigonometric considerations to demonstrate that the above parameterization of the cycloid is correct. Describe a geometric reinterpretation of t (not as merely time), and then, presuming t is time in seconds, determine the linear horizontal speed of the rolling disk's center. Presume that x, y and R are expressed in meters. In three dimensions, one can describe curves in space parametrically by specifying a parameter and three functions defined on some common domain (which may be a proper subset of the natural domains of any of the given functions). That is, one may describe a curve C to be the locus of points determined parametrically by the triple of equations x(t) = f(t) ; y(t) = g(t) ; z(t) = h(t) ; t 2 D ⊆ R : The curve C is connected if and only if D is a connected subset of R and all three functions are continuous on any open subinterval of D. 3 We've already encountered a simple example: parametric equations to describe lines in R . In particular, if f, g, and h, are all linear functions of t, then the corresponding parametric equations describe a line if D = R and a line segment if D = [a; b] is an interval. We already know that we can organize this same data using a vector whose components are the three functions f, g, and h. Exercise 1.3. : Give a geometric argument that the line segment from (x0; y0; z0) to (x1; y1; z1) may be parameterized by x(t) = (1 − t)x0 + tx1 ; y(t) = (1 − t)y0 + ty1 ; z(t) = (1 − t)z0 + tz1 ; t 2 [0; 1] : Give a new parameterization for a time parameter t (in seconds) that describes a particle smoothly traveling from (x0; y0; z0) to (x1; y1; z1) and back to (x0; y0; z0) in as many seconds as the twice length of the line segment. 1.2. Vector Valued Functions. Definition. Given a domain D ⊂ R, a (3D) vector valued function is a vector valued map r : D ! 3 R , described by specifying functions f; g; h : D ! R: r(t) = f(t)^ı + g(t)^ + h(t) k^ = hf(t); g(t); h(t)i ; t 2 D: 2 One can specify a vector valued function with values in R by omitting the third component; more generally one can define vector valued functions with n-dimensional vector values by choosing n functions f1; : : : ; fn : D ! R as the components of an n-vector r(t) = hf1(t); : : : ; fn(t)i. The domain 3 2 n of r, D, is often denoted Dom(r). The image or range of r is the set of points in R (or R , or R ) whose position vectors are values of r(t) for some t. Note that the components must all be defined on a common domain. One can ask about natural domain, as one does for single variable functions: for a given r described by component formulae, what is the largest domain in R on which r is defined? 2 2/21/20 Multivariate Calculus: Curves Havens Proposition. The natural domain of a vector valued function is the intersection of the natural domains of the components. Exercise 1.4. : Prove the above proposition. A vector valued function is equivalent to parametric equations, but provides us a neat geometric language to work with curves. Indeed, if the domain is connected and the component functions have (locally) connected graphs (meaning no jumps/skips or undefined points), then the image of 2 3 a two or three dimensional vector valued function is a curve in R or R respectively. Example. Describe and visualize the curve given parametrically by r(t) = cos(t)^ı + sin(t)^ + t k^. Solution: First, consider the \shadow" (projection) of this parameterization on the xy-plane. Clearly, this is a unit circle, traversed counterclockwise. Indeed, one can easily verify that the x and y components of r satisfy x2 + y2 = 1. Thus, the whole curve is in fact confined to this cylinder. The z-component tells the rest of the story: as t increases, the height above the xy-plane increases commensurately. Thus, this curve is a circular helix. Figure 2. The circular helix parameterized by r(t) = cos(t)^ı + sin(t)^ + t k^. Exercise 1.5. Recall that conic sections have cartesian equations of the following form: x2 y2 x2 y2 Ellipse: + = 1 ; Parabola: y = ax2 ; Hyperbola: − = ±1 : a2 b2 a2 b2 For each, give a vector valued function parameterizing the curves with a domain of R. Exercise 1.6. Give Cartesian equations for the following plane curves: 2 4 (a.) The curve which is the image of the vector valued function r(t) = t ^ı + t ^, t 2 R. (b.) The trajectory of a particle whose position after t seconds is r(t) = hsin3(πt); 1 − sin2(πt)i, 0 ≤ t ≤ 1. (c.) The \witch of Agnesi," a curve named for Maria Agnesi, author of the first three-term calculus text, given by the vector valued function r(t) = 2a cot(θ)^ı + 2a sin2(θ)^, t 2 [0; 2π], where a is a positive real constant. 3 2/21/20 Multivariate Calculus: Curves Havens Figure 3. The Witch of Agnesi, \witch" being a mistranslation of the Italian word versiera (perhaps for the 'versine' function 1 − cos t.) Whereas in the plane, a single equation involving both variables may implicitly specify a curve (such as a conic section), curves in 3-space often arise as the intersection of a pair of surfaces. Indeed, conic sections are realized as intersections of a double cone with a plane, hence their name. But non-planar curves arise, e.g. as the intersection loci of cylinders or other surfaces. Example. Sketch the curve of intersection of the parabolic cylinder y = x2 and the cubic cylinder z = x3, and parameterize this curve. Solution: y = x2 gives a parabolic cylinder whose traces look like the familiar parabola translated into planes of constant z, while the cubic cylinder appears as the standard cubic sketched in the xz-plane, then translated into planes of constant y.